Theory for the bending anisotropy of lipid membranes and tubule formation

We study the spontaneous symmetry breaking of the bending rigidity of lipid membranes in two principal directions by a Landau mean-field theory. When the temperature is below the tilting temperature [Formula Presented], the coupling between molecular orientation and membrane local curvature square leads to an increase of the bending rigidity in the tilting direction and therefore a spontaneous symmetry breaking in two principal directions. The asymmetry [Formula Presented] of the bending rigidity undergoes a continuous change upon cooling and grows as [Formula Presented] for [Formula Presented] We discuss this anisotropical effect on the tilt structure of the ripple phase [Formula Presented] of a nearly flat membrane and the sphere-to-tubule transition in a dilute solution of lipids. The transition between the spherical vesicle phase and the tubular phase is predicted to be first order.